Short course "Motivic Cohomology". Lecture 1: An introduction to higher Chow groups and triangulated categories of motives

Lecture 1/3.

This will be a short course giving an introduction to the parallel theories of motivic cohomology furnished by Bloch’s higher Chow groups and Voevodsky’s motivic cohomology. Along the way, we will introduce a number of motivic categories and describe how they give a framework for motivic cohomology. We will say a bit about two applications of the theory: constructions of the “motivic” spectral sequence from motivic cohomology to algebraic K-theory, and the proof of the Bloch-Kato conjectures relating mod n-motivic cohomology with mod n étale cohomology. Finally, we will describe theories of motivic cohomology over a general base-scheme.

1. Lecture 1: An introduction to higher Chow groups and triangulated categories of motives In the first lecture, we will introduce Bloch’s cycle complex, and Bloch’s higher Chow groups. After detailing the construction, we will briefly describe the basic properties of the higher Chow groups as a Bloch-Ogus twisted duality theory: functoriality, homotopy invariance, projective bundle formula, and the crucial localization sequence. We conclude with an introduction to the relation of the higher Chow group with algebraic K-theory via the Chern character and the Bloch-Lichtenbaum spectral sequence.

In the second part of this lecture, we will introduces some categories of motives. We start with Grothendieck’s category of Chow motives for smooth projective varieties. Next, we discuss Voevodsky’s triangulated category of geometric motives and his sheaf-theoretic version, the triangulated category of effective motives. We conclude with Voevodsky’s embedding theorem and the categorical description of Suslin homology